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Landing in Binary Asteroids: A Global Map of Feasible Descend Opportunities for Unpowered Spacecraft

J.P. Sáncheza*, O. Celikb

a Space Research Group, Centre of Autonomous and Cyber-Physical Systems, Cranfield University,UK.jp.sanchez@cranfield.ac.ukb Department of Space and Astronautical Sciences, The Graduate University for Advanced Studies (SOKENDAI),Sagamihara, Japan. onur.celik@ac.jaxa.jp.* Corresponding Author

AbstractAsteroid surface science provides the necessary “ground-truth” to validate and enhance remote sensing from

orbiting spacecraft. Yet, due to uncertainties associated with the dynamical environment near asteroids, it isgenerally prudent for the main spacecraft to remain at a safe distance. Instead, small landers could be used muchmore daringly. This paper explores the potential for ballistic landing opportunities in binary asteroid systems. Thedynamics near a binary asteroid are modelled by means of the Circular Restricted Three Body Problem, whichprovides a reasonable representation of a standard binary system. Natural landing trajectories are sought that allowfor deployment from safe distances and touchdown with minimum local-vertical velocity. The necessary coefficientof restitution to ensure a successful landing and the effects of navigation and deployment errors are also analysed.Assuming deployment errors in the order of 10 meters and 1 cm/s (1-sigma), the results show that ballistic descentlanding operations are likely to be successful if targeting near equatorial regions with longitude within 320o to 20o inthe secondary of the binary system.

Keywords: Binary asteroid missions, trajectory design, ballistic descent trajectories, circular restricted three bodyproblem, deployment errors and covariance matrix.

1. IntroductionNear Earth asteroids (NEAs) are the easiest celestial

objects to reach from Earth (excl. the Moon), and offer aunique window to the early stages of accretion anddifferentiation of the inner planets of our Solar System.As such, they have become appealing targets for sciencemissions. For such missions, on-board remote sensinginstrumentation is paramount, however, in-situmeasurements provide the necessary “ground-truth” toenhance the science return.

The chaoticity of the strongly irregular dynamicalenvironment found at asteroids, however, entails somedaring challenges for navigation, and thus, mostmissions spend long periods of times (~months)stationed well beyond the asteroid’s Hill radius, wherethe heliocentric dynamical environment is stillpredominant, and thus much more easily predicted.

NanoSats and other shoebox-sized landers havealready been identified as potential valuable assets forin-situ asteroid exploration, since, due to their low cost,they can be used much more daringly. However, due toconstraints in mass and volume, these systems may onlyallow for extremely crude orbit and landing control.This paper thus explores the potential for passivelanding opportunities that may be enabled by theasteroid’s natural dynamics.

Particularly, the paper focuses on binary asteroidsystems, i.e. asteroids with a satellite, which arebelieved to account for about 15% of the NEA

population. The dynamics near a binary asteroid arethen modelled by means of the Circular RestrictedThree Body Problem, which provides a reasonablyrepresentative model for a standard binary system.Natural landing trajectories are sought that allow for adeployment well outside the orbit of the asteroid’ssatellite, and a touchdown with minimum local-verticalvelocity.

By taking a purely deterministic approach, the paperprovides a global map of the minimum touchdownvelocity, as well as the required coefficient of restitution(i.e. energy damping) in order to ensure that thespacecraft would not bounce away from the system.Similarly, the deployment velocities from a mothershipstationed outside the orbit of the satellite can also becomputed.

The functionality and reliability of these trajectoriesis next discussed by generating landing conditions for aspecific coefficient of restitution and modellingnavigation and deployment errors. The covariancematrices for a global set of landing conditions can thenbe propagated to the surface and the most robust landinglocations are identified.

The remaining of the paper is organized through thefollowing sections: Section 2 describes the motivationand state-of-the-art of asteroid landing; Section 3 verybriefly describes the methodology of the deterministicapproach to design landing trajectories; Section 4introduces the guidance, navigation and control

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challenges for landing operations of passive systems;and finally Section 5 provides some conclusions.

2. State-of-the-artIn so far, a total of eight spacecraft have flown to, or

by, twelve asteroids. However, hundreds of asteroidswill need to be visited, if we are to fully understand theworkings of our Solar System; particularly, itscompositional diversity. Even more limited is thenumber of surface probes that have touched down on anasteroid or comet surface. Two landers have so farattempted to touchdown with limited success:MINERVA1 and Philae2. While instead three mainspacecraft have performed similar landing operations;Hayabusa3, NEAR-Shoemaker4 and Rosetta5. Although,only Hayabusa attempted such a manoeuvre before theend of mission phase.

Landing on a small body, such as an asteroid orcomet, substantially differs from landing on a deepergravity well, such as that of Mars or the Moon. Theextremely weak gravitational environment found insmall bodies makes purely ballistic descent trajectoriesa viable option, since the touchdown velocities can besafely managed by the landing gear. All theaforementioned landings have in fact attempted ballisticdescents. However, the same weak gravitationalenvironment entails a completely different challenge:Unless sufficient energy is damped at touchdown, thelander may well bounce away of the asteroid, or bounceinto a badly illuminated conditions, which mayseriously jeopardize the mission6.

Particularly, this paper focuses on landingopportunities for binary asteroids. These are systems oftwo asteroids orbiting around their common centre ofmass7. The only visited binary asteroid so far is Ida-Dactyl8, which was flown by Galileo spacecraft at 12.4km/s, on the 26th August 1993, with a close approach of~2400 km9.

The planetary science community has however a realinterest in returning to a binary, particularly with arendezvous mission. Such a mission would clearlycontribute to settle the debate on the formation of thesesystems. Moreover, the poles of the larger asteroid (i.e.the primary thereafter) are likely to feature freshmaterial10, and thus unweathered and pristine. Hence,missions have been proposed and studied; such asMarcoPolo-R11 and BASIX12. Apart from theirfascinating geological and geophysical history, binaryasteroids offer also the opportunity to investigate theeffectiveness of future asteroid deflection techniques:The NASA/ESA co-operative AIDA mission wasintended to impact the small asteroid (i.e. the secondarythereafter) in the binary asteroid Didymos, in order totest the kinetic impact deflection technique13.

AIDA consisted of two distinct spacecraft; NASA’sDART (Double Asteroid Redirection Test), which is toimpact the Didymos’ secondary and ESA’s Asteroid

Impact Mission (AIM), which was to rendezvous withDidymos to observe DART’s impact. The AIM proposalincluded a small MASCOT-2 lander, of about 12 kg, aswell as two 3U CubeSats to be deployed in theneighbourhood of Didymos. Even if AIM’s futureappears unlikely, since 2016 ESA Ministerial, theexamples above indicate a clear interest to land smallunpowered payloads into the surface of a binary system.

However, navigating within a binary asteroid systemposes serious challenges, due to its highly perturbeddynamics and its natural satellite. Therefore, it mightnot be the best solution to put a mothership at risk bygetting it into the vicinity of the binary in order todeploy an unpowered lander. The natural dynamics inbinary asteroids could instead be exploited to achieveballistic landing from a safe distance.

Ballistic landing on binary asteroids by means ofnatural manifold trajectories were studied by Tardiveland Scheeres14. A follow up work also discussed thisstrategy within the context of the deployment of a lightlander as an optional payload for MarcoPolo-Rmission15, including an statistical analysis ofuncertainties for deployments near the L2 point. Asimilar L2 release strategy is depicted in Ferrari andLavagna16. The consequent motion of the lander on theasteroid surface, after the initial touchdown, has alsobeen investigated by several studies17-19.

This paper particularly focuses on searching landingopportunities for soft local-vertical touchdown in abinary asteroid. Landing trajectories are reverseengineered from the surface of both objects (i.e. primaryand secondary), by propagating the model backwards intime. A dense grid of latitude-longitude nodes,homogenously distributed over the entire surface ofboth objects, allows us to obtain a global map offeasible descend opportunities for unpowered landers.

3. A Global Map of Landing OpportunitiesThe mission architecture considered in this paper is

that of a mothership which carries one (or several)landers. Once the mothership reaches the binary system,it would start orbiting near the binary and perform thenecessary science operations. The mothership should besufficiently close to the binary system to performsatisfactorily detailed observations; however, it shouldalso remain sufficiently far away from the system’sbarycentre to prevent any collision risk or potentialcontamination of instruments and sensors.

The following section will briefly summarize theasteroid model and dynamical framework used. Then,Section 3.2 provides a general description of themethodology used to map the local vertical landingconditions over the entire surface of the binary. Section3.3 will discuss some of the results. For further detailand/or discussion, the reader is referred to Celik andSanchez20.

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3.1 Asteroid Model and Dynamical FrameworkBinary asteroids are considered here as being of

spherical shape and of constant density; both primaryand secondary objects. Whereas all the secondaries thathave been observed are somewhat elongated, primariesfeature an approximately spherical shape21-23.Nevertheless, a spherical shape provides a reasonablerepresentation, which captures well the main underlyingfeatures of the dynamics of the full three body problem.Besides, the methodology described in Section 3.2would also be applicable to higher fidelity shapemodels.

Hence, the Circular Restricted Three Body Problem(CR3BP) is used as a dynamical framework to estimatethe motion of a massless spacecraft near the twoasteroids. Thus, in a synodic reference frame centred inthe barycentre of the binary system, the unpropelledmotion of the landing spacecraft can be modelled by24,25

with the usual normalized set of ordinary differentialequations:

2

2

x yx

y xy

zy

∂Ω= +

∂

∂Ω= − +

∂

∂Ω=

∂

(1)

where the potential function Ω is defined as: 2 2 1

2 p s

x y

r r

µ µ+ −Ω = + + (2)

where rp and rs are the distances to the primary andsecondary respectively. The mass parameter µ can beconveniently defined as:

µn

n=

+

3

3 1(3)

where n is the secondary-to-primary radius ratio of thespecific binary we want to model20.

The mechanical energy in the CR3BP is representedby the Jacobi Constant C:

C U V= − 22 (4)where V is the spacecraft speed given in rotating frameof reference.

For a given set of initial conditions, the JacobiConstant defines the accessible and forbidden zones ofmotion. From the expression in Eq.(4), it can bededuced that there are locations, for which at someenergies, the velocity of the motion may become animaginary number, which clearly indicates the physicalimpossibility of motion within those locations at thoseenergy levels. Hence, zero-velocity surfaces (ZVS) aredefined when one considers the condition V=0. ZVSshows which regions are accessible to motion at a givenenergy.

Figure 1 depicts a scenario on which the mothershipis flying on a sufficiently high Jacobi Constant such thatmotion can only occur around the primary, thesecondary or in a region exterior to the whole binarysystem. Such an operational orbit would be ideal for thesafety of the mothership, since transitions between thesethree regions of allowed motion cannot occur as thereare no connections between them. Thus, the mothershipcannot possibly collide with the asteroid.

Figure 1. Illustration of mission architecture

3.2 MethodologyAccording to the scenario depicted in Figure 1, the

deployment must be performed such that the lander isprovided with sufficient energy for the ZVS to open atthe L2 point, and a transition may thus occur. It isenvisaged that the mothership is equipped with a simplespring mechanism capable to deploy the lander at mostat 2 m/s26.

Once the scenario is defined, landing trajectories arenow in fact reverse engineered, by defining the point inthe surface where we want to land, and propagatingbackwards until some satisfactory deployment conditionis found. Note that once the landing site is defined, onlyone design parameter requires to be determined, sincelocal vertical direction is assumed, and this is thetouchdown speed vl.

In backward propagation, a touchdown speed vl suchthat its Jacobi Constant is equal to the L2 point vl(CL2)would be ensured not to reach the exterior region, wherethe mothership lingers. On the other hand, a touchdownspeed vl>> vl(CL2) will instead easily reach the exteriorregion. A bisection search algorithm is then used toiterate between these two limits, until a minimumvelocity is found that reaches the orbital altitude wherethe mothership may be encountered.

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3.3 ResultsA key requirement for landing opportunities is to

allow for the slowest possible touchdown speed; firstly,to prevent damage to the spacecraft and its payload and,secondly, to minimize the risk of bouncing off thesystem if sufficient damping of the energy at touchdownis not achieved.

Table 1. The properties of notional averagebinary asteroid20.

Primary SecondaryDiameter [m] 1000 250

Density [kg/m3] 1700Mass [kg] 8.9 x 1011 1.4 x 1010

Orbit semi-major axis [m] 1950Orbital period [h] 19.4 h

Figure 2 summarises the results of the bisectionsearch to find the minimum possible landing velocity.The results have been dimensionalized to represent anaverage binary system20 such as that represented inTable 1.

Figure 2. Minimum touchdown speed for thebinary system in Table 1. Landing speed for theSecondary object is also represented in a Mercatorprojection in the top part of the figure.

As shown in Figure 2, the primary occupies thelowest part of the gravity well of the binary. Thus, asexpected, the ballistic descent trajectories touchdown athigher velocities than at the secondary. Most of thesurface of the secondary is instead reachable with touchdown velocities near 10 cm/s. It is noteworthy that mostof the secondary surface is accessible at touchdownvelocities lower than the two-body escape velocity,approximately 35 cm/sec in this case.

Figure 3 shows several examples of landingtrajectories, both plotted in the synodic reference frameof the dynamical framework used, and also in the

inertial reference frame. Thus, for example, we see howthe secondary landing at latitude ϕ and longitude λ of 0 degrees is clearly influence by the dynamics near the L2libration point. This, however, represents a nearlycircular orbit that moves parallelly with the secondary,as seen in the inertial reference frame.

Figure 3. Landing trajectory for three locationswithin the binary, as seen in the synodic reference

frame and the inertial reference frame.

Note that the secondary is assumed to be tidallylocked21, hence, its attitude can be assumed fixed in thesynodic reference frame, and the body’s prime meridian(λ=0) can be arbitrarily defined as on the x-axis and on the far side of the system (facing the classical L2 point).However, the primary will have a fast rotation, which islikely to be near the point at which a body withnegligible cohesion starts shedding mass21. The notionalaverage binary used here assumes that the rotation ofthe primary is 2.2h, as well as a rotational axis is alsoperpendicular to the orbital plane of the secondary21.

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Once the minimum velocities have been computed,the coefficient of restitution, or energy damping,required for these landing conditions to settle onto thesurface of the asteroid, can be identified. This paperconsiders a simple model for the interaction of thespacecraft with the asteroid’s surface; that of a bouncingball with a specified coefficient of restitution ε20,27.

The touchdown conditions can be described in local-vertical (LV) components as:

( )ˆ ˆ ˆLV lv−= ⋅ =v n v n n (5)

where the vector n corresponds to the normal to thesurface at the contact point, and the superscript (-) refersto the conditions just an infinitesimal instant before therebound. Considering an impulsive rebound attouchdown with some loss of energy due to, forexample, the deformation of the surface and/orspacecraft, the velocity conditions an infinitesimalinstant after the rebound can be approximated as:

ˆLV lvε+= −v n (6)

where the coefficient of restitution ε defines thedissipation of energy.

Figure 4 summarises the values of the coefficient ofrestitution ε that ensures a sufficient condition to remainpermanently in the neighbourhood of the body wherethe landing has been attempted. This sufficientcondition is computed by estimating a coefficient of

restitution11 LVC LVL Closeε + −= v v , where

1LVC

+v represents

the local vertical velocity after touchdown such that theJacobi constant of the bouncing trajectory is that of theL1 equilibrium. This energy limit after the first bounceis chosen so that no escape can possibly occur from theneighbourhood of the asteroid, since zero velocitysurfaces will be closed both at L1 and L2 points.

Figure 4. Required coefficient of restitution ε toensure ZVS closed at L1 and L2 points.

Figure 4 provides important insights into thefeasibility of the ballistic landing in a binary system. Itshall be noted that Hayabusa and Philae’s touchdownmeasured coefficients of restitution of ε <0.852,3. Hence,considering the available data from past missions, it isclear that if assuming a conservative estimate for thecoefficients of restitution (ε ~0.9), only the far side of the secondary would be available for landing. However,appropriate structural design may well allow forcoefficients of restitution ε near ~0.628. At ε slightly lower than 0.7, large equatorial regions in the primarybecome available as landing locations.

Philae landing at comet 67P2 observed coefficientsof restitution ε≈0.7 in multiple bounces. Thus, the remaining of the paper will assume as unfeasibleregions for landing any location where the coefficientsof restitution ε in Figure 4 is below this limit. The primary is thus discarded and we will focus on thesecondary as a landing location. Nevertheless, as shownin Celik and Sanchez20, the results in Figure 4 are likelyto be only worse case estimates of the necessarycoefficient of restitution, since multiple bounces arelikely to occur, and thus multiple opportunities energydamping before the spacecraft may definitely escape thesystem.

3.4 DeploymentLet us now focus on the deployment operation. The

mothership is likely to release the landing system whilein a trajectory taking it near to the binary, but yet safeaccording to the ZVS discussion in section 3.1. Thus,we assume that the release trajectory has a periapsis atthe deployment point, and apoapsis near the sphere ofinfluence of the binary system (SOI). At the deploymentpoint the mothership shall have a normalized velocity

scv :

2 2 ˆsc release

release release SOI

rr r r

= − − +

v θ

where ˆ ˆ ˆ= ×θ h r , r is the release position unit radius

vector and h is the direction of the ballistic descenttrajectory momentum vector. The initial state vector of

the ballistic descent [ ]release release

r v was computed with

the aforementioned bisection algorithm20. Note that the

state vector [ ]release release

r v is chosen such that two

constraints are satisfied:• The duration of the descent trajectory must be

less than 12-h.• The distance to the barycenter of the binary must

be larger than 2340 meters.

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The latter constrain ensures that the minimumdistance to the secondary is 1.25 larger than the distanceto the L2 point.

The deployment spring mechanism in themothership must then provide an impulse such as:

spring release sc= −v v v .

Note that, ignoring navigation errors, the release

location releaser is assumed to coincide with the position

of the mothership at the release time scr .

According the above deployment model, a relativelyreduced region of the secondary is available for landingat coefficient of restitution ε>0.7. This region is depicted in Figure 5. It can be noted that some regionsin the far side are no longer reachable, due to the factthat the ballistic descent trajectory takes more than 12hours for those locations. This however could be solvedby allowing touchdown velocities larger than theabsolute minimum touchdown velocity (Figure 2), aswill be seen later.

The homogeneous color in the Mercator projectorindicates that any possible landing location requires adeployment velocity smaller than 20 cm/s, with aminimum deployment velocity of ~2 cm/s. Note that thedeployment mechanism of Rosetta spacecraft wascapable of providing a separation speed between 5 – 50cm/s29. More relevant to these results, AIM’sdeployment mechanism was being designed to provide2-5 cm/s within ±1 cm/s accuracy30.

Figure 5. Deployment velocity as a function oflocation in the surface of the Secondary.

4. GNC ChallengesThe purely deterministic analysis shown in the

previous section demonstrated that ballistic descenttrajectories are certainly possible in large regions of thesecondary and possible for the primary. Although forthe primary, a landing system capable to ensuresufficient energy damping at touchdown may benecessary.

However, inherent to the benefits and advantages oflow energy trajectories also come their instabilities31.Moreover, the deployment will also be affected byinaccuracies of the knowledge of the state vector of themothership, as well as by errors during the deploymentcaused by the deployment mechanism.

Many other sources of error and perturbations exist;such as the mass distribution of the asteroid. This paper,however, focuses now only on GNC relatedinaccuracies and their effect on the trajectories of thelander. Previous work by the authors also considered thevariability of the asteroid’s density, and it was shown tohave a limited effect32. Finally, the fact that sphericalshape is assumed may not necessarily be considered as asource of error; since if the shape of the asteroid wasknown, the same strategy could be used to compute newballistic landing opportunities.4.1 Deployment operation and errors

The navigation model will assume that both position

scr and velocity scv of the mothership will be affected

by navigation errors, and that the deployment velocity

springv may also be affected due to inaccuracies of the

deployment mechanism. Table 2 summarizes the errorsin position and velocity caused by the GNC and springmechanism.

Table 2. Uncertainty and Error Sources

3σ value GNC position accuracy ±30 mGNC velocity accuracy ±3 cm/sSpring magnitude error ±30%

Spring angle error ±15o

Table 2 can be translated into a diagonal covariancematrix Q such as:

2

2

2

2

2

2

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

x

y

z

x

y

z

v

v

v

Q

σ

σ

σ

σ

σ

σ

=

where the diagonal contain the square root of thestandard deviation for each component of the statevector. Q can then be propagated to the asteroid surfaceby means of the transition matrix Φ of the landingreference trajectory. The covariance matrix at thelanding time Q(tl) can thus be computed as:

( ) ( )( ) , ( ) ,Tl l d d l dQ t t t Q t t t= Φ Φ

where the subscripts l and d denote landing anddeployment times, respectively.

The covariance matrix at the landing time Q(tl)provides a linear approximation of the sensitivity of thelanding trajectory to errors in GNC and deployment.The position errors in Q(tl) are represented within thefirst 3x3 submatrix of Q(tl):

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( )

l l lxx xy xz

l l lyx yy yz TR

l l lzx zy zz

l

BL BR

Q Q Q

Q Q Q

Q Q QQ t

=

Q

Q Q

However, the position errors will be best representedin a topocentric reference frame using the principal axis,in such a way that the previous top left matrix can berepresented in its diagonal form:

( )

2

2

2

0 0

0 0

0 0l

a

Q t b

c

=

At this point, the semi-major and minor axis (a, b)represent the footprint of the 1-sigma Gaussiandistribution of the deployment errors. Finally, this datacan be best represented with a single figure of merit A2σ

such as:

2 24

s

abA

rσ =

where rs is the radius of the secondary.A2σ represents the area of the 2-sigma distribution

footprint in units of cross sectional area of thesecondary. Thus, a 2-sigma distribution footprint A2σ

equal to 1 would represent a footprint as large as theasteroid itself, thus indicating a highly unreliabledeployment. One would thus ideally aim fordeployments such that A2σ<<1.4.2 Landing at minimum possible touchdown velocity

Figure 6 summarizes the 2-sigma distributionfootprint A2σ as a function of latitude ϕ and longitude λ over the surface of the secondary. The fact that onlysmall regions feature values of A2σ<1 indicates that atthe accuracies in navigation and deployment assumed inTable 2 minimum touchdown velocity descenttrajectories are not robust enough to provide a reliablelanding.

Figure 6. 2-sigma distribution footprint A2σ as afunction of latitude ϕ and longitude λ over thesurface of the secondary.

It follows then that landing trajectories with largertouchdown speed should be attempted. A larger than theminimum possible touchdown velocity implies a muchfaster descent trajectory, and thus shorter landingoperations. Hence, initial errors in the descent trajectorymay have less time to propagate, and thus may have alesser impact on the descent trajectory. Nevertheless,note that the spring error is proportional to the velocitymagnitude, and thus the latter statement requires to bedemonstrated.4.3 Landing at coefficients of restitution of 0.7

As discussed in section 3.3, a coefficient ofrestitution ε of 0.7 is defined as the minimum allowed restitution. Hence, landing operations that matchprecisely this coefficient of restitution are computed, asdescribed in section 3.4. Figure 7 shows now therobustness of these descent trajectories to the sameerrors during the deployment described in Table 2. Notethat the color scale has now changed since now most ofthe available landing space present values of A2σ below1. Particularly, a relatively large region with A2σ in therange of [0.1, 0.2] extends longitudes from 310o to 25o

and latitudes extending beyond 70o.The minimum value of A2σ occurs at a longitude λ

of 355o and the slightly off-equatorial latitude ϕ of 2o.The A2σ value at this location is of 0.114. A MonteCarlo analysis of this landing spot, with 10,000randomly generated landing conditions, certifies that theprobability of a first touchdown is of 100%. If we thenassume a coefficient of restitution of 0.7 or lower, wecan then be confident that the lander will remain in thesurface of the secondary.

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Figure 7. Adimensional Area of the 2-sigma ellipsoid of uncertainty of the landing state.

5. ConclusionThis paper has investigated the possibility to

benefit from the natural dynamics near a binaryasteroid as an efficient mean to deliver a lander, orscientific payload, onto its surface. The results showthat landing at the primary requires some minimumlanding system capable to damp excess of energy, toavoid bouncing out of the asteroid.

However, landing on the secondary seemscompatible with completely passive systems, even ifconsidering navigation and deployment errors in theorder of 10 meters and 1 cm/s (1-sigma distribution).Near equatorial landing, in the region extending from320o to 20o longitude, appear to be sufficiently robustto deployment errors.

AcknowledgementsOnur Celik is supported by Japanese Government

Research Scholarship (Monbukagakusho) for his PhDresearch. Onur Celik also acknowledges the RoyalObservatory of Belgium for providing the short-termresearch grant to carry out part of this study.

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